I feel also air field (wind), height field (land), light field (photons), and so on.

My point exactly - fields are real, they are not merely mathematical models. (It's just that some words acquire a more precise meaning, being formalized as part of a model, and while it's true that models may contain an additional "scaffolding" that has no analogue in reality, field is not one of those.)

Field is mathematical abstraction, used in mathematical model, to represent a physical thing.

Physical things are real. Mathematical abstractions are not. OpenGL is not real too, but it looks very real and accurately predicts reality. In OpenGL, field is array, e.g. "float[][][] field;".

That’s like saying, “Cat is a word used in a language to represent a physical thing.”

No, it's like saying that nature is built on kind of vertex buffer, because OpenGL accurately predicts reality.